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  "body": "\n## Proof\n\nWe apply the orbit-stabiliser theorem to the [[action by conjugation|0P8CY9V437]], $g \\cdot x = g x g^{-1}$.\n\n1. The [[orbit|00F2JFA41J]] of $a$ under conjugation is $\\set{g a g^{-1} : g \\in G}$. By the [[definition of a conjugacy class|0Q88KQD17A]], this is $\\operatorname{Cl}(a)$.\n2. The [[stabiliser|0RGGF8ZS6D]] of $a$ is $\\set{g \\in G : g a g^{-1} = a}$. Multiplying on the right by $g$, or by $g^{-1}$, shows $g a g^{-1} = a$ if and only if $g a = a g$.\n3. So the stabiliser of $a$ is $\\set{g \\in G : ga = ag}$, which is the [[centraliser|0Q0Z1BEXTF]] $C_G(a)$.\n4. By the [[orbit-stabiliser theorem|0E2HSCKZ6C]] and steps 1 and 3, $\\abs{\\operatorname{Cl}(a)} = [G : C_G(a)]$.\n5. If $G$ is finite, the [[orbit size formula|0S19CWFZM2]] shows that the size of each orbit divides $\\abs{G}$. By step 1, $\\abs{\\operatorname{Cl}(a)}$ divides $\\abs{G}$.\n\n## Idea\n\nConjugation is an action, its orbits are the conjugacy classes, and the elements that fix $a$ are those that commute with $a$. The orbit-stabiliser theorem then turns the class size into the index of the centraliser.\n",
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